Cell complexes obtained from sets with relations
Takahiro Matsushita · arXiv (Cornell University) · 2014
For a positive integer $r$, an $r$-set we say in this paper is a set $X$ with a subset of $X^r$. For $r$-sets $X$ and $Y$, we construct the poset ${\rm Hom}(X,Y)$ called the Hom complex, and the simplicial set ${\rm Sing}(X,Y)$ called the singular complex. We show that their geometric realizations are homotopy equivalent. The Hom complex of $r$-sets is the generalization of the Hom complexes of graphs which have been applied to the graph coloring problem in combinatorics. On the other hand, singular complexes are more compatible with categorical construction of $r$-sets than Hom complexes. By using these complexes, the theory of beat points of posets, the $\times$-homotopy theory of graphs established by Dochtermann, and the strong homotopy theory of finite simplicial complexes established by Barmak and Minian can be unified to the case of $r$-sets.